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 A019318 Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same. 5
 1, 2, 16, 252, 6814, 244344, 10746377, 553319048, 32611596056, 2163792255680, 159593799888052, 12952412056879996, 1147044793316531040, 110066314584030859544, 11375695977099383509351, 1259843950257390597789296, 148842380543159458506703546, 18685311541775061906510072648, 2483858381692984848273972297368, 348545122958862200122401771463328 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of n X n binary matrices with n ones under action of dihedral group of the square D_4. LINKS Mario Velucchi, Title? Mario Velucchi, Different Dispositions in the ChessBoard. FORMULA See Velucchi link or the PARI program. Note that the polynomial whose coefficient of a^k is divided by 8 differs based upon whether the term's index is even or odd. Let A(n) = C(n^2, n); B(n) = C((n^2-(n mod 2))/2, n/2); C(n) = C((n^2-(n mod 2))/4, n/4); D(n) = Sum(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p)). Then a(n) = (A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4), (A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4), (A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4), (A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4). - David W. Wilson, May 29 2003 EXAMPLE For n=3 the 16 solutions are 111 110 110 110 110 110 110 101 101 101 100 100 100 010 010 010 000 100 010 001 000 000 000 010 000 000 011 010 001 110 101 010 000 000 000 000 100 010 001 000 100 010 000 001 010 000 000 010 MATHEMATICA p[a_, b_, n_] := If[EvenQ[n], (a+b)^(n^2) + 2*(a+b)^n*(a^2 + b^2)^((n^2 - n)/2) + 3*(a^2 + b^2)^(n^2/2) + 2*(a^4 + b^4)^(n^2/4), (a+b)^(n^2) + 2*(a+b)*(a^4 + b^4)^((n^2-1)/4) + (a+b)*(a^2 + b^2)^((n^2-1)/2) + 4*(a+b)^n*(a^2 + b^2)^((n^2-n)/2)]; Table[Coefficient[p[a, 1, k], a, k]/8, {k, 1, 20}] (* Jean-François Alcover, Nov 12 2013, translated from Pari *) PROG (PARI) {p(a, b, N) = if(N%2==0, (a+b)^(N^2) + 2*(a+b)^N*(a^2+b^2)^((N^2-N)/2) + 3*(a^2+b^2)^(N^2/2) + 2*(a^4+b^4)^(N^2/4), (a+b)^(N^2) + 2*(a+b)*(a^4+b^4)^((N^2-1)/4) + (a+b)*(a^2+b^2)^((N^2-1)/2) + 4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))} for(k=1, 20, print1(polcoeff(p(a, 1, k), k)/8, ", ")) CROSSREFS Cf. A054252 and A014409. Sequence in context: A138764 A009833 A009044 * A090727 A108242 A140307 Adjacent sequences:  A019315 A019316 A019317 * A019319 A019320 A019321 KEYWORD nonn,nice AUTHOR Mario Velucchi (mathchess(AT)velucchi.it) EXTENSIONS More terms from Rick L. Shepherd and David W. Wilson, May 28 2003 STATUS approved

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Last modified November 18 21:04 EST 2018. Contains 317331 sequences. (Running on oeis4.)